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Linear Algebra

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Inner Product Space: Inner Products Coming From Matrices

LINALG-4WUGFN

Let $\langle\ ,\ \rangle : \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^3$ be defined as:

$$\langle \boldsymbol{u}, \boldsymbol{v}\rangle = (A\boldsymbol{u})^T(A\boldsymbol{v})$$

$\ldots$ for $\boldsymbol{u},\ \boldsymbol{v} \in \mathbb{R}^3$ and for some $3\times 3$ matrix $A$. Which of the following matrices make $\langle\ ,\ \rangle$ into an inner product.

Select ALL that apply.

A

$\begin{bmatrix} 0 & 0 & 0 \\\ 0 & 0 & 0 \\\ 0 & 0 & 0 \end{bmatrix}$

B

$\begin{bmatrix} 1 & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 1 \end{bmatrix}$

C

$\begin{bmatrix} 2 & 0 & 0 \\\ 0 & 2 & 0 \\\ 0 & 0 & 2 \end{bmatrix}$

D

$\begin{bmatrix} 1 & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 0 \end{bmatrix}$

E

$\begin{bmatrix} 1 & 2 & 3 \\\ 3 & 2 & 1 \\\ 2 & 1 & 3 \end{bmatrix}$

F

$\begin{bmatrix} 1 & -2 & 3 \\\ 3 & -2 & 1 \\\ -2 & 1 & 3 \end{bmatrix}$